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Gaussian Mixture Model GMM is based on Gaussian probability density function, which can smoothly approximate density distribution of arbitrary shape. Since GMM has multiple models and its fine division characteristics, it can be used for complex object modeling.

Suppose there is a batch of observation data ${X\text{ }=\text{ }{ \{ {x\mathop{{}}\nolimits_{{1}},x\mathop{{}}\nolimits_{{2}},…,x\mathop{{}}\nolimits_{{n}}} \} }}$ , and its distribution in d-dimensional space is not ellipsoidal, then it is not suitable to be described by a single Gaussian density. If all points are generated by a single Gaussian distribution, by mixing data points with different distributions together, this distribution method is a Gaussian mixture distribution.

From a mathematical point of view, the probability distribution density function of the data can be expressed by a weighting function:

${p{ \left( {x\mathop{{}}\nolimits_{{i}}} \right) }\text{ }=\text{ }{\mathop{ \sum }\nolimits_{{j=1}}^{{M}}{a\mathop{{}}\nolimits_{{j}}N\mathop{{}}\nolimits_{{j}}{ \left( {x\mathop{{}}\nolimits_{{i}}; \mu \mathop{{}}\nolimits_{{j}}, \Sigma \mathop{{}}\nolimits_{{j}}} \right) }}}}$

Among them ${{\mathop{ \sum }\nolimits_{{j=1}}^{{M}}{a\mathop{{}}\nolimits_{{j}}}} \text{ }=\text{ } 1}$ , and ${N\mathop{{}}\nolimits_{{j}}{ \left( {x; \mu \mathop{{}}\nolimits_{{j}}, \Sigma \mathop{{}}\nolimits_{{j}}} \right) }\text{ }=\text{ }\frac{{1}}{{\sqrt{{{ \left( {2 \pi } \right) }\mathop{{}}\nolimits^{{m}}{ \left| { \Sigma \mathop{{}}\nolimits_{{j}}} \right| }}}}}exp{ \left[ {-\frac{{1}}{{2}}{ \left( {x\text{ }-\text{ } \mu \mathop{{}}\nolimits_{{j}}} \right) }\mathop{{}}\nolimits^{{T}} \Sigma \mathop{{}}\nolimits_{{j}}\mathop{{}}\nolimits^{{-1}}{ \left( {x\text{ }-\text{ } \mu \mathop{{}}\nolimits_{{j}}} \right) }} \right] }}$ represents the mixture function model of the j-th single Gaussian function.

Theoretically, GMM can fit any type of distribution and is usually used to solve the problem of multiple different distributions in the same set.